order-4 octagonal tiling
{{short description|Regular tiling of the hyperbolic plane}}
{{Uniform hyperbolic tiles db|Reg hyperbolic tiling stat table|U84_0}}
In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}. Its checkerboard coloring can be called a octaoctagonal tiling, and Schläfli symbol of r{8,8}.
Uniform constructions
There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,8] kaleidoscope. Removing the mirror between the order 2 and 4 points, [8,8,1+], gives [(8,8,4)], (*884) symmetry. Removing two mirrors as [8,4*], leaves remaining mirrors *4444 symmetry.
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|[8,4] |[8,8] |[(8,4,8)] = [8,8,1+] {{CDD|node_c1|8|node_h0|4|node_c2}} = {{CDD|label4|branch_c1|2a2b-cross|nodeab_c2}} |[1+,8,8,1+] |
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!Symbol |{8,4} |r{8,8} |r(8,4,8) = r{8,8}{{frac|1|2}} |r{8,4}{{frac|1|8}} = r{8,8}{{frac|1|4}} |
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|{{CDD|node_1|8|node|4|node}} |{{CDD|node|8|node_1|8|node}} |{{CDD|node|8|node_1|8|node_h0}} = {{CDD|node|split1-88|branch_11|label4}} {{CDD|node_1|8|node_h0|4|node}} = {{CDD|label4|branch_11|2a2b-cross|nodes}} |{{CDD|node_h0|8|node_1|8|node_h0}} = {{CDD|labelh|node|split1-88|branch_11|label4}} = |
Symmetry
This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting as edges of a regular hexagon. This symmetry by orbifold notation is called (*22222222) or (*28) with 8 order-2 mirror intersections. In Coxeter notation can be represented as [8*,4], removing two of three mirrors (passing through the octagon center) in the [8,4] symmetry. Adding a bisecting mirror through 2 vertices of an octagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 4 bisecting mirrors through the vertices defines *444 symmetry. Adding 4 bisecting mirrors through the edge defines *4222 symmetry. Adding all 8 bisectors leads to full *842 symmetry.
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The kaleidoscopic domains can be seen as bicolored octagonal tiling, representing mirror images of the fundamental domain. This coloring represents the uniform tiling r{8,8}, a quasiregular tiling and it can be called a octaoctagonal tiling.
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Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram {{CDD|node_1|8|node|n|node}}, progressing to infinity.
{{Order-4_regular_tilings}}
{{Order-8 regular tilings}}
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram {{CDD|node_1|n|node|4|node}}, with n progressing to infinity.
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{{Order 8-4 tiling table}}
{{Order 8-8 tiling table}}
See also
{{Commons category|Order-4 octagonal tiling}}
References
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, {{isbn|978-1-56881-220-5}} (Chapter 19, The Hyperbolic Archimedean Tessellations)
- {{Cite book|title=The Beauty of Geometry: Twelve Essays|year=1999|publisher=Dover Publications|lccn=99035678|isbn=0-486-40919-8|chapter=Chapter 10: Regular honeycombs in hyperbolic space}}
External links
- {{MathWorld | urlname= HyperbolicTiling | title = Hyperbolic tiling}}
- {{MathWorld | urlname=PoincareHyperbolicDisk | title = Poincaré hyperbolic disk }}
- [http://bork.hampshire.edu/~bernie/hyper/ Hyperbolic and Spherical Tiling Gallery]
- [http://geometrygames.org/KaleidoTile/index.html KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings]
- [http://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch]
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